the Doppler effect (classical)

The Doppler effect is noth­ing real­ly new. It’s well-known that Doppler fre­quen­cy shift occurs for sound as well as light; there are, how­ev­er, some sub­tle dif­fer­ences between the two, owing main­ly to Einstein’s spe­cial the­o­ry of rel­a­tiv­i­ty. This arti­cle describes the clas­si­cal Doppler effect as it occurs e.g. for sound waves and in which the involved veloc­i­ties are low enough to be able to neglect rel­a­tivis­tic effects. The rel­a­tivis­tic ver­sion of the Doppler effect as it relates to e.g. elec­tro­mag­net­ic waves will be the top­ic of a lat­er arti­cle.

The Doppler effect is described in var­i­ous lev­els of detail in a mul­ti­tude of places [1–3]. What I couldn’t find on the ‘net, though, is a clear math­e­mat­i­cal deriva­tion of it by means of plane wave func­tions in the form

$$\begin{aligned}
E(\mathbf{r},t) & =\frac{1}{2}\Bigl(A_{0}\exp\bigl(i\omega t-i\mathbf{k}\cdot\mathbf{r}\bigr)+\mathrm{c.c.}\Bigr)\\
& =\Re\Bigl\lbrace A_{0}\exp\bigl(i\omega t-i\mathbf{k}\cdot\mathbf{r}\bigr)\Bigr\rbrace\\
& =\Bigl|A_{0}\Bigr|\cos\Bigl(\omega t-\mathbf{k}\cdot\mathbf{r}+\mathrm{arg}\bigl[A_{0}\bigr]\Bigr)\label{plane-wave}
\end{aligned}$$

as I reg­u­lar­ly use them here (e.g. in this arti­cle) and wher­ev­er else they might come in handy. To be as gen­er­al as pos­si­ble with­out being bound to any par­tic­u­lar coor­di­nate sys­tem, these plane waves should be giv­en in terms of the posi­tion vec­tor $\mathbf{r}$ and the rec­i­p­ro­cal wave vec­tor $\mathbf{k}$, describ­ing a spa­tial fre­quen­cy with direc­tion.

Plane Waves

Just about any field dis­tri­b­u­tion can be decom­posed into plane waves in order to math­e­mat­i­cal­ly prop­a­gate them from here to there, as has been done e.g. in the arti­cle on the angu­lar spec­trum. So if we can describe a lin­ear effect in terms of the plane wave we can describe it for any oth­er wave by decom­pos­ing that wave into plane waves, cal­cu­lat­ing the effect on each of the plane waves and super­pos­ing the plane wave solu­tions to obtain the desired result. Note that this super­po­si­tion will only work for lin­ear effects / sys­tems.
Now, $E$ in $\eqref{plane-wave}$ can rep­re­sent the real-val­ued elec­tric field in the case of light prop­a­ga­tion or air pres­sure in the case of sound waves, $A_{0}$ is the (con­stant) com­plex ampli­tude which con­tains both mag­ni­tude and phase infor­ma­tion of the source wave, $\omega$ is the oscil­la­tion fre­quen­cy of the source, $\mathbf{k}$ is the afore­men­tioned wave vec­tor, and “c.c.” stands for the com­plex con­ju­gate of every­thing that pre­cedes it. The term

$$A(t)=A_{0}\exp\bigl(i\omega t-i\mathbf{k}\cdot\mathbf{r}\bigr)\label{plane-wave-complex}$$

is called the pha­sor and ful­ly describes the wave. We will use this pha­sor a lot for lin­ear sys­tems, because it allows us to work with the rel­a­tive­ly sim­ple expo­nen­tial func­tions instead of the more com­pli­cat­ed cosine. Both are equal­ly valid, though.

The fre­quen­cy $\omega^\prime$ that the receiv­er at posi­tion $\mathbf{r}$ detects is sim­ply the time deriv­a­tive of the phase of the pha­sor $A(t)$:

$$\begin{aligned}
\omega^\prime & =\partial_{t}\arg\bigl[A(\mathbf{r},t)\bigr] \label{frequency}\\
& =\omega
\end{aligned}$$

when the source ampli­tude $A_{0}$ is con­stant. As long as noth­ing moves, the receiv­er sees (or hears) the same fre­quen­cy that was sent. When things get mov­ing, how­ev­er, this changes as a result of the Doppler effect.

There are actu­al­ly two slight­ly dif­fer­ent Doppler effects in the clas­si­cal domain. When we have a trans­port­ing medi­um like we have with sound waves, it makes a dif­fer­ence whether the wave source is sta­tion­ary with respect to the medi­um (and only the receiv­er moves) or whether the source is mov­ing, because the veloc­i­ty $c$ of the waves is defined rel­a­tive to the medi­um. Both cas­es can also be com­bined in var­i­ous ways.

Moving Receiver

The math­e­mat­i­cal­ly most straight­for­ward case is the mov­ing receiv­er with a sta­tion­ary source: The vec­tor $\mathbf{r}$ in $\eqref{plane-wave}$ describes the posi­tion of the receiv­er in space (the source is assumed to be at the ori­gin of the coor­di­nate sys­tem). The motion of the receiv­er is described by

$$\mathbf{r}=\mathbf{r}_{0}+\mathbf{v}_{R}t=\mathbf{r}_{0}+v_{R}\mathbf{\hat{v}}_{R}t$$

in which $v_{R}=\bigl|\mathbf{v}_{R}\bigr|$ is its veloc­i­ty, $\mathbf{\hat{v}}_{R}$ is a unit vec­tor in the direc­tion of motion and $\mathbf{r}_{0}$ is its ini­tial posi­tion. For the mov­ing receiv­er, $\eqref{plane-wave}$ becomes

$$A_{R}(\mathbf{r},t)=A_{0}\exp\Bigl[i\omega t-i\mathbf{k}\cdot\bigl(\mathbf{r}_{0}+\mathbf{v}_{R}t\bigr)\Bigr]\label{plane-wave-R}$$

The received fre­quen­cy is deter­mined anal­o­gous to $\eqref{frequency}$:

$$\begin{aligned}
\omega_{R}^\prime & =\partial_{t}\arg\bigl[A_{R}(\mathbf{r},t)\bigr]\\
& =\omega-\mathbf{k}\cdot\mathbf{v}_{R}\\
& =\omega\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{R}}{\omega}\biggr)\\
& =\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)
\end{aligned}\label{frequency-R}$$

where we used

$$\mathbf{k}=\mathbf{\hat{k}}\frac{\omega}{c}\label{normalized-wave-vector}$$

Hence, there is a fre­quen­cy shift which depends on the direc­tion of the receiv­er motion rel­a­tive to the direc­tion of the wave com­ing from the source. If the receiv­er approach­es the source direct­ly ($\mathbf{k}$ is antipar­al­lel to $\mathbf{v}$), the fre­quen­cy increase is max­i­mum. The observed fre­quen­cy shift due sole­ly to the change of the received phase as a result of the receiv­er motion with­in the field. Nei­ther the spa­tial dis­tri­b­u­tion nor the oscil­la­tion fre­quen­cy of the field has changed.

Also note that the case $\omega\rightarrow\infty$ (describ­ing a son­ic boom) is not achiev­able since the receiv­er veloc­i­ty can­not reach infin­i­ty.

Moving Source

Things change a bit with a source on the move. We could try and switch our frame of ref­er­ence to the mov­ing source to re-cre­ate the sit­u­a­tion with a mov­ing receiv­er and a sta­tion­ary source in the new coor­di­nates, apply $\eqref{plane-wave-R}$ and $\eqref{frequency-R}$, and be done with it. In this case we would dis­re­gard, how­ev­er, that the medi­um is now also mov­ing with respect to our coor­di­nate sys­tem. And since the wave veloc­i­ty $c$ is defined rel­a­tive to the medi­um, our result would be wrong.

To cor­rect this error, we need to fig­ure out the new wave vec­tor $\mathbf{k}_{S}$ in the new frame of ref­er­ence in which the medi­um is mov­ing. We’ll start by decom­pos­ing the medi­um motion into a motion par­al­lel to the ini­tial wave vec­tor $\mathbf{k}$ and one orthog­o­nal to it. The mag­ni­tude of the for­mer is

$$\Delta c=-\mathbf{\hat{k}}\cdot\mathbf{v}_{S}=-\frac{c}{\omega}\mathbf{k}\cdot\mathbf{v}_{S}$$

in which $\mathbf{v}_{S}$ is the veloc­i­ty vec­tor of the source in the ref­er­ence frame in which the medi­um is at rest and hence $-\mathbf{v}_{S}$ is the veloc­i­ty vec­tor of the medi­um in the mov­ing frame. This com­po­nent of the medi­um veloc­i­ty direct­ly alters the phase veloc­i­ty $c$ of the wave since the phase veloc­i­ty is defined rel­a­tive to it. The oth­er com­po­nent, orthog­o­nal to $\mathbf{k}$ and par­al­lel to the wave fronts, will shift the wave side­ways, but since the plane wave is infi­nite­ly extend­ed in that direc­tion, such a shift caus­es no observ­able change and can thus be neglect­ed. Since the only change is in the direc­tion $\mathbf{\hat{k}}$ of the wave vec­tor, that direc­tion does not change, mere­ly its mag­ni­tude is affect­ed.

Then, with the mod­i­fied phase veloc­i­ty

$$c^\prime=c+\Delta c=c\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega}\biggr)\label{velocity-S}$$

we have the new wave vec­tor

$$\begin{aligned}
\mathbf{k}_{S} & =\frac{\omega}{c^\prime}\mathbf{\hat{k}}\\
& =\mathbf{k}\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{S}}{c}\biggr)^{-1}\\
& =\mathbf{k}\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega}\biggr)^{-1}
\end{aligned}\label{wave-vector-S}$$

As our frame of ref­er­ence trans­for­ma­tion is pure­ly trans­la­tion­al, the same wave vec­tor $\mathbf{k}_{S}$ is observed in the mov­ing as well as the fixed frame. Note that a sim­ple addi­tion of veloc­i­ties as in $\eqref{velocity-S}$ is only pos­si­ble in clas­si­cal physics. To deter­mine the received fre­quen­cy as before, we’ll remain in the mov­ing frame, not­ing that in this frame the receiv­er also moves with $-\mathbf{v}_{S}$ as does the medi­um. We insert $\eqref{wave-vector-S}$ into $\eqref{plane-wave-complex}$,

$$A_{S}(\mathbf{r},t)=A_{0}\exp\Bigl[i\omega t-i\mathbf{k}_{S}\cdot\bigl(\mathbf{r}_{0}-\mathbf{v}_{S}t\bigr)\Bigr]$$

and deter­mine $\omega_{S}^\prime$ anal­o­gous to $\eqref{frequency}$:

$$\begin{aligned}
\omega_{S}^\prime & =\partial_{t}\arg\bigl[A_{S}(\mathbf{r},t)\bigr]\\
& =\omega+\mathbf{k}\cdot\mathbf{v}_{S}\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega}\biggr)^{-1}\\
& =\omega\biggl(1+\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega-\mathbf{k}\cdot\mathbf{v}_{S}}\biggr)\\
& =\omega\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega}\biggr)^{-1}\\
& =\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{S}}{c}\biggr)^{-1}\label{frequency-S}
\end{aligned}$$

Note that if the com­po­nent of the source veloc­i­ty par­al­lel to the wave vec­tor equals the wave veloc­i­ty $c$ in the medi­um, both $\mathbf{k}$ and $\omega_{S}^\prime$ become infi­nite, result­ing in a son­ic boom in the case of sound waves. Also, the appar­ent fre­quen­cy shift is now due the receiv­er motion with­in the wave field as above, as well as the change of the wave­length with­in the wave field. The fre­quen­cy of the wave again remains the same.

Both Moving

When both source and receiv­er are mov­ing, we only need to alter $\eqref{plane-wave-S}$ to addi­tion­al­ly account for the receiv­er motion:

$$A_{RS}(\mathbf{r},t)=A_{0}\exp\Bigl[i\omega t-i\mathbf{k}_{S}\cdot\bigl(\mathbf{r}_{0}-\mathbf{v}_{S}t+\mathbf{v}_{R}t\bigr)\Bigr]$$

result­ing in a received fre­quen­cy of

$$\begin{aligned}
\omega_{RS}^\prime & =\omega\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{R}}{\omega}\biggr)\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{S}}{\omega}\biggr)^{-1}\\
& =\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{S}}{c}\biggr)^{-1}
\end{aligned}$$

The cor­rec­tion fac­tor to the fre­quen­cy $\omega$ in this case is the prod­uct of the cor­rec­tion fac­tors in $\eqref{frequency-R}$ and $\eqref{frequency-S}$. Also, if both source and receiv­er move with the same veloc­i­ty rel­a­tive to the medi­um, the received fre­quen­cy is again equal to the source fre­quen­cy $\omega$.

Note that the fre­quen­cy $\omega_{RS}^\prime$ depends on the absolute veloc­i­ties of source and receiv­er, not just how they move rel­a­tive to each oth­er. This is due to the cru­cial role of the medi­um in clas­si­cal physics.

Echo from a Moving Reflector

A sta­tion­ary source emits a wave which is reflect­ed back towards the source by a mov­ing receiv­er. A Doppler shift as above occurs in both direc­tions. The fre­quen­cy observed by the source can again be found by a com­bi­na­tion of the first two cas­es. The out­go­ing wave is received by the reflec­tor with an observed fre­quen­cy of $\eqref{frequency-R}$,

$$\omega_{SRS}^\prime=\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)$$

The reflec­tor in turn becomes a mov­ing source which emits waves with the mod­i­fied fre­quen­cy $\omega_{SRS}^\prime$ which is observed by it and also a mod­i­fied wave vec­tor cor­re­spond­ing to that fre­quen­cy

$$\mathbf{k}^\prime=-\mathbf{k}\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)$$

cf. also $\eqref{normalized-wave-vector}$. The minus sign results from the change of direc­tion upon reflec­tion. The fre­quen­cy observed by the source upon recep­tion of the waves emit­ted by the reflec­tor is giv­en by $\eqref{frequency-S}$ in which $\mathbf{v}_{R}=\mathbf{v}_{S}$ (both describ­ing the motion of the reflec­tor) and $\omega$ and $\mathbf{k}$ have been replaced by $\omega_{SRS}^\prime$ and $\mathbf{k}^\prime$, respec­tive­ly:

$$\begin{aligned}
\omega_{SRS}^{\prime\prime} & =\omega_{SRS}^\prime\biggl(1-\frac{\mathbf{k}^\prime\cdot\mathbf{v}_{R}}{\omega_{SRS}^\prime}\biggr)^{-1}\\
& =\omega\biggl(1-\frac{\mathbf{k}\cdot\mathbf{v}_{R}}{\omega}\biggr)\biggl(1+\frac{\mathbf{k}\cdot\mathbf{v}_{R}}{\omega}\biggr)^{-1}\\
& =\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)\biggl(1+\frac{\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c}\biggr)^{-1}\\
& =\omega\frac{c-\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}{c+\mathbf{\hat{k}}\cdot\mathbf{v}_{R}}
\end{aligned}
\label{frequency-SRS}$$

Decomposition

Note that the per­ceived Doppler shift in $\eqref{frequency-SRS}$ depends sole­ly on the com­po­nent of the reflec­tor motion which is par­al­lel to the prop­a­ga­tion veloc­i­ty of the plane wave. If the source is not a source of plane waves, which it can­not be for plane waves are infi­nite­ly extend­ed, we can decom­pose what­ev­er waves the source does emit into an angu­lar spec­trum of plane waves, as men­tioned above. See also the arti­cle on the angu­lar spec­trum in the con­text of Fouri­er dif­frac­tion. Each of these plane waves has a dif­fer­ent direc­tion $\mathbf{\hat{k}}$ and thus will gen­er­al­ly cause a dif­fer­ent detect­ed fre­quen­cy shift upon reflec­tion and sub­se­quent recep­tion. The source will thus see (or hear) a whole spec­trum of reflect­ed fre­quen­cies.

This is com­pound­ed by the finite extent of the receiv­er and thus a finite cross-sec­tion for reflec­tion. Each incom­ing plane wave is spa­tial­ly fil­tered upon reflec­tion so that the reflect­ed por­tion of it must again be decom­posed into its angu­lar spec­trum, each com­po­nent of which is received with a dif­fer­ent Doppler shift back at the source. The reflec­tion spec­trum can pos­si­bly be decon­vo­lut­ed and ana­lyzed to reveal infor­ma­tion about the reflec­tion cross-sec­tion, but such an analy­sis exceeds the scope of this arti­cle.


[1] Doppler effect [Wikipedia]
[2] Doppler effect [James B. Calvert]
[3] Doppler Shift for Sound and Light [Reflec­tions on Rel­a­tiv­i­ty]


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